Advertising adstock is the carry-over effect of some advertisement to a consumer over time. Finding the decay rate or half-life of advertising is a common question of interest to many advertisers to determine how effective advertising builds the awareness of their brand, and how that awareness decays over time.

Adstock is traditionally applied to advertisement via TV, and models are used to determine the best-fitting adstock rate of TV to Sales, or some sort of outcome (i.e. awareness). However in most cases, one would have other mediums such as Radio, Print, Digital, Social, etc.

I took the Nonlinear Least Squares approach to solving for the optimal adstock rate commonly applied to a single advertising medium, and augmented it to take in multiple variables.

My motivation for developing this multivariate approach is that modeling adstock rates for each advertisement medium independently may not be sufficient given that multiple mediums affect the outcome, and need to be accounted for collectively.

Method

Let   = adstock rate, and  = error at time i. Then we can model Sales (or some outcome of advertising) as:

where

Now let’s say there are three advertising mediums that we want to compute adstock rates for. In this multivariate scenario, this model would look like this:

The goal is to find the optimal rates of all  values, using Nonlinear Least Squares. The intercept that is computed from the model can also be interpreted as the Base, or the base level of sales or outcome if there were no advertising at all.

Example in R

For this example I generated a sample data with 3 ad variables (each representing some advertisement medium) with 104 obervations (representing roughly 2 years of weekly data). Then sales is generated from base + ad variables w/ ad stocking, with added random noise.

FYI: If you aren’t using pacman already, it is a great package management tool and I would highly recommend it (link to Github).

# generate sample data
pacman::p_load(minpack.lm)
set.seed(2222)

# adstock function
adstock<-function(x,rate=0){
  return(as.numeric(stats::filter(x=x,filter=rate,method="recursive")))
}

# generate base (intercept) + noise, and random values for ad1, ad2, and ad3
n_weeks = 104
base = 50
ad1 = sapply(rnorm(n_weeks, mean = 20, sd = 10), function(x) round(max(x, 0), 0))
ad2 = sapply(rnorm(n_weeks, mean = 20, sd = 10), function(x) round(max(x, 0), 0))
ad3 = sapply(rnorm(n_weeks, mean = 20, sd = 10), function(x) round(max(x, 0), 0))

# adstock rates
ad1_rate = .7
ad2_rate = .4
ad3_rate = .5

# generate sales data from the base + ad vairables w/ ad stocking, with random noise
sales = round(base + adstock(ad1, ad1_rate) + adstock(ad2, ad2_rate) + adstock(ad3, ad3_rate) + rnorm(n_weeks, sd = 5), 0)

I wrote a Multivariate Adstock Function in R, with special thanks to Angela Ju, whose code from this article I adopted and augmented. The equation from is implemented in the R function using the nls function to fit a nonlinear least squares with the adstock function.

This function can take a data.frame of any number of column(s) (or advertisement mediums), and will calculate the optimal adstock rate for each column in the input data.

#multivariate adstock function
AdstockRateMV <- function(Impact, Ads, maxiter = 100){
  # parameters
  params = letters[2:(ncol(Ads)+1)]
  # ad variables
  ads = paste0("ad_", params)
  # rate variables
  rates = paste0("rate_", params)
  # create partial formula
  param_fm = paste(
    paste(params, "*adstock(", ads,  ",", rates, ")", sep = ""),
    collapse = " + "
  )
  # create whole formula
  fm = as.formula(paste("Impact ~ a +", param_fm))
  # starting values for nls
  start = c(rep(1, length(params) + 1), rep(.1, length(rates)))
  names(start) = c("a", params, rates)
  # input data
  Ads_df = Ads
  names(Ads_df) = ads
  Data = cbind(Impact, Ads_df)
  # fit model
  modFit  0)){
    library(minpack.lm)
    lower = c(rep(-Inf, length(params) + 1), rep(0, length(rates)))
    upper = c(rep(Inf, length(params) + 1), rep(1, length(rates)))
    modFit <- nlsLM(fm, data = Data, start = start,
                   lower = lower, upper = upper,
                   control = nls.lm.control(maxiter = maxiter))
  }
  # model coefficients
  AdstockInt = round(summary(modFit)$coefficients[1, 1])
  AdstockCoef = round(summary(modFit)$coefficients[params, 1], 2)
  AdstockRate = round(summary(modFit)$coefficients[rates, 1], 2)
  # print formula with coefficients
  param_fm_coefs = paste(
    paste(round(AdstockCoef, 2), " * adstock(", names(Ads),  ", ", round(AdstockRate, 2), ")", sep = ""),
    collapse = " + "
  )
  fm_coefs = as.formula(paste("Impact ~ ", AdstockInt, " +", param_fm_coefs))
  # rename rates with original variable names
  names(AdstockRate) = paste0("rate_", names(Ads))
  # calculate percent error
  mape = mean(abs((Impact-predict(modFit))/Impact) * 100)
  # return outputs
  return(list(fm = fm_coefs, base = AdstockInt, rates = AdstockRate, mape = mape))
}

The function takes in an Impact (a vector or single-column data frame of some advertising outcome), Ads (data frame of advertisement variables), and maxiter (maximum # of iterations for convergence), and returns the adstock model formula, base value, adstock rate for each ads considered, and the Mean average percent error (MAPE) between the predicted outcome and actual outcome.

First as a baseline, let’s fit a univariate model for adstock rates for each advertisement mediums.

# adstock for ad1
Impact = sales
(mod = AdstockRateMV(Impact, data.frame(ad1)))
## $fm
## Impact ~ 106 + 0.96 * adstock(ad1, 0.78)
##
##
## $base
## [1] 106
##
## $rates
## rate_ad1
##     0.78
##
## $mape
## [1] 6.9729

For Ad 1, the model estimates base as 106 and adstock rate as 0.78.

# adstock for ad2
(mod = AdstockRateMV(Impact, data.frame(ad2)))
## $fm
## Impact ~ 137 + 1.05 * adstock(ad2, 0.59)
##
##
## $base
## [1] 137
##
## $rates
## rate_ad2
##     0.59
##
## $mape
## [1] 8.064316

For Ad 2, the model estimates base as 137 and adstock rate as 0.59.

# adstock for ad3
(mod = AdstockRateMV(Impact, data.frame(ad3)))
## $fm
## Impact ~ 130 + 1.17 * adstock(ad3, 0.61)
##
##
## $base
## [1] 130
##
## $rates
## rate_ad3
##     0.61
##
## $mape
## [1] 7.768505

For Ad 3, the model estimates base as 130 and adstock rate as 0.61.

However, the original parameters used to simulate the data are base of 50 with rates of 0.7, 0.4, 0.5. To my previous point, modeling adstock for each medium independently may not be sufficient due to omitted-variable bias, and thus should be considered together.

Let us now compute the adstock rates for all three advertisement variables together in a multivariate model.

# multivariate adstock model
Ads = data.frame(ad1, ad2, ad3 )
(mod = AdstockRateMV(Impact, Ads))
## $fm
## Impact ~ 51 + 0.98 * adstock(ad1, 0.7) + 1.02 * adstock(ad2,
##     0.37) + 0.97 * adstock(ad3, 0.53)
##
##
## $base
## [1] 51
##
## $rates
## rate_ad1 rate_ad2 rate_ad3
##     0.70     0.37     0.53
##
## $mape
## [1] 2.160336

The model estimates base as 51 and adstock rates as 0.7, 0.37, 0.53. With a MAPE of 2.16%, and in comparison to base of 50 and rates of 0.7, 0.4, 0.5, this is a fairly accurate estimate.

Simulation

Now let’s do a simulation with n = 100 random samples taken from normal distributions.

# simulation
adstock_sim = function(){
  # generate base (intercept) + noise, and random values for ad1, ad2, and ad3
  base = 50
  ad1 = sapply(rnorm(n_weeks, mean = 20, sd = 10), function(x) round(max(x, 0), 0))
  ad2 = sapply(rnorm(n_weeks, mean = 20, sd = 10), function(x) round(max(x, 0), 0))
  ad3 = sapply(rnorm(n_weeks, mean = 20, sd = 10), function(x) round(max(x, 0), 0))
  # adstock rates
  ad1_rate = .7
  ad2_rate = .4
  ad3_rate = .5
  # generate sales data from the base + ad vairables w/ ad stocking, with random noise
  sales = round(base + adstock(ad1, ad1_rate) + adstock(ad2, ad2_rate) + adstock(ad3, ad3_rate) + rnorm(n_weeks, sd = 5), 0)
  # fit model
  Impact = sales
  Ads = data.frame(ad1, ad2, ad3 )
  mod = AdstockRateMV(Impact, Ads)
  return(c(base = mod[[2]], mod[[3]], mape = mod[[4]]))
}

# replicate 100 times
mod_rep = replicate(n = 100, adstock_sim())
rowMeans(mod_rep)
##      base  rate_ad1  rate_ad2  rate_ad3      mape
## 50.180000  0.699000  0.400900  0.492400  2.099492

With a simulation of 100 samples, the model estimates the average base as 50 and average rates as 0.7, 0.4, 0.5, with a mean MAPE of 2.1%.

The caveat here is that simulations can be built to produce any results as expected (and is certainly the case here), but in practice, I believe this multivariate approach to adstock modeling provides a better representation of adstock rates of different advertisment mediums, compared to a univariate approach.

If you liked this post, please check out my other posts at https://justrthings.com/.